Suppose that candidatesAandBare in an election.Areceivesavotes andBreceivesbvotes, witha > b.

The Ballot Problem:
How many ways can thea + bvotes be arranged so thatAmaintains a lead throughout the counting of
the ballots?

Answer: .

The Generalized Ballot
Problem:Supposea
> kbfor some positive integerk.How many ways can thea +
bvotes be arranged so thatAalways has more thanktimes as many votes asBthroughout the counting of the ballots?

Answer: .

(Some authors refer to both of the above as the ballot
problem, and use “generalized ballot problem” for something even more general.)

Please note: Many
excellent articles and reputable sources claim (incorrectly) that André used
the reflection method.I have created
this list not to question the quality of the sources, but only to show the
prevalence of the misattribution.Sources are listed starting with the most recent.

I.P.
Goulden and Luis G. Serrano, Maintaining the Spirit of the Reflection
Principle when the Boundary has Arbitrary Integer Slope, J. Combinatorial Theory (A) 104
(2003) 317-326.
p. 2: “André gave a direct geometric bijection between the subset of bad
paths and the set A of all paths
from (1, -1) to (m, n), and the result then follows
immediately, since |A| = C(m + n, m – 1). In the bijection, the
initial portion of the path up to the first point that lies on the line y = x – 1 is reflected about the line y = x – 1, and so
André’s beautiful method of proof is called the reflection principle.”

J.H.
Van Lint and R.M. Wilson, A Course
in Combinatorics, Cambridge University Press, 2001.
p. 151: “The reflection principle of Fig. 14.2 was used by the French
combinatorialist D. André (1840-1917) in his solution of Bertrand’s famous
ballot problem…”

I.
Karatzas and S.E. Shreve,Brownian Motion and Stochastic Calculus,
Springer, 1998.
When introducing the reflection principle, they cite P. Lévy 1948 p. 293
(see below).Then they write “Here
is the argument of Désiré André…” and proceed with the reflection method.

H.
Bauer, Probability Theory,
Walter de Gruyter, Berlin, New York, 1996.
p. 231: “In the literature, this reflection principle is usually
attributed to D. André (1840-1918).It occurs in the form of such a geometric argument in André
[1887].”

I don’t believe Comtet actually means to attribute the reflection method
to André, though he may be unclear on this matter.After posing the ballot problem he
writes, “This is the famous ballot
problem formulated by [Bertrand, 1887]; we give the elegant solution
of [André, 1887].”He then goes on
to write “We first formulate the principle
of reflection, which essentially is due to André.”Comtet includes references to many of
André’s works.His use of the word “essentially”
seems to indicate that he recognizes a subtle distinction.

On page 505 Papoulis writes “The reflection principle† (of Désiré André),
loosely phrased, says that the functions of a Wiener-Lévy process that
cross a line w = d (this line
will be denoted by Ld)
continue on symmetrical paths.”At
the dagger, Papoulis includes a footnote citing J.L. Doob, Stochastic Processes (see below).

W.
Feller, An Introduction to
Probability Theory and Its Applications, Second Edition, John Wiley
& Sons, Inc., New York 1957.

p. 66.Feller introduces the ballot
problem then writes in the footnote: “For the history and literature see
A. Dvoretzky and T. Motzkin, A problem of arrangements, Duke Mathematical
Journal, vol. 14 (1947), pp. 305-313.As these authors point out, most of the formally different proofs
in reality use the reflection principle (lemma 1 of section 2), but
without the geometric interpretation this principle loses its simplicity
and appears as a curious trick.”

p. 70.Upon considering the
reflection method, Feller writes in the footnote “The probability
literature attributes this method to D. André (1887).The text reduces it to a lemma on random
walks.The classical difference
equations of random walks (chapter XIV) closely resemble differential
equations, and the reflection principle (even a stronger form of it) is
familiar in that theory under the name of Lord Kelvin’s method of images.”

In the first edition of this book (1950) no mention is made of André or
the reflection principle.

On page 393, in a section on Brownian movement process, Doob writes “The
exact evaluation (2.3) and similar exact evaluations are easily made, once
sample function continuity has been proved, using what is known as the
reflection principle of Désiré André.”

In the preface, Doob thanks Feller for reading and commenting on earlier
versions of the manuscript.Perhaps
this is why Feller includes the reflection principle in the second edition
of his book, but not the first?

In the book Brownian Motion and
Stochastic Calculus by Shreve and Karatzas (Springer 1998), page 293
of this Lévy source is cited.However, I’ve been unable to find any reference to André or the
reflection principle in Lévy.

Are
there any references to André and the reflection principle prior to 1953?